Use the unit circle to evaluate the function.

Use the unit circle to evaluate the function. Tan 7pie/6

The Correct Answer and Explanation is:

To evaluate tan(7π/6) using the unit circle:

✅ Correct Answer:

\tan\left(\frac{7\pi}{6}\right) = \frac{1}{\sqrt{3}} \cdot (-1)^2 = \frac{1}{\sqrt{3}} \cdot 1 = \frac{1}{\sqrt{3}} \text{ with the sign } (-), \text{ so } \boxed{\tan\left(\frac{7\pi}{6}\right) = \frac{1}{\sqrt{3}} \cdot (-1) = \boxed{\frac{1}{\sqrt{3}}(-1) = \boxed{\frac{1}{\sqrt{3}} = -\frac{\sqrt{3}}{3}}}

🧠 Step-by-Step Explanation:

1. Understand the angle: 7π6 is in radians. To understand its location on the unit circle, convert it to degrees:\frac{7\pi}{6} \text{ is in radians. To understand its location on the unit circle, convert it to degrees:}67π​ is in radians. To understand its location on the unit circle, convert it to degrees: 7π6⋅180∘π=210∘\frac{7\pi}{6} \cdot \frac{180^\circ}{\pi} = 210^\circ67π​⋅π180∘​=210∘
2. Find the reference angle:
   The reference angle is the acute angle formed with the x-axis. Since 210° is in the third quadrant: 210∘−180∘=30∘210^\circ – 180^\circ = 30^\circ210∘−180∘=30∘ So the reference angle is 30° (or π/6 radians).
3. Use known values from the unit circle:
   For the reference angle π/6:
   • sin⁡(π/6)=12\sin(\pi/6) = \frac{1}{2}sin(π/6)=21​
   • cos⁡(π/6)=32\cos(\pi/6) = \frac{\sqrt{3}}{2}cos(π/6)=23​​
   • tan⁡(π/6)=13=33\tan(\pi/6) = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}tan(π/6)=3​1​=33​​
4. Determine the sign of tangent in the third quadrant:
   In the third quadrant, both sine and cosine are negative, and since tangent is sine divided by cosine, the negatives cancel out: tan⁡(7π6)=−12−32=13=33\tan\left(\frac{7\pi}{6}\right) = \frac{-\frac{1}{2}}{-\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}tan(67π​)=−23​​−21​​=3​1​=33​​
5. Final step – correct sign:
   Since both sine and cosine are negative in quadrant III, their ratio is positive. So: tan⁡(7π6)=33\boxed{\tan\left(\frac{7\pi}{6}\right) = \frac{\sqrt{3}}{3}}tan(67π​)=33​​​

However, there is a sign mistake in the earlier step. Let’s correct:

At 210°, sine and cosine are both negative:

• sin⁡(210∘)=−12\sin(210^\circ) = -\frac{1}{2}sin(210∘)=−21​
• cos⁡(210∘)=−32\cos(210^\circ) = -\frac{\sqrt{3}}{2}cos(210∘)=−23​​

So:tan⁡(210∘)=−12−32=13=33\tan(210^\circ) = \frac{-\frac{1}{2}}{-\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}tan(210∘)=−23​​−21​​=3​1​=33​​

Therefore, the final correct answer is:tan⁡(7π6)=33\boxed{\tan\left(\frac{7\pi}{6}\right) = \frac{\sqrt{3}}{3}}tan(67π​)=33​​​